Here, each BAN wore one transmitter/on-body sensor on Right–Hip (RH). The sampling rate for each channel in this dataset is 10 Hz. Although, there were 20 coexisting BANs, all the B2B links were not used to estimate the results. In fact the result is averaged over around 160 links for the body-to-body (RH–RH) channel.
5.3
Test of Significance for WSS
To investigate the wide-sense-stationarity (WSS) of the B2B channels, we use the ‘frequentist’ approach along with the null hypothesis significance testing (NHST)
5.3 Test of Significance for WSS 115
Table 5.1: The body-to-body channels with different sensor locations. ‘×’ indicates the B2B channels correspond to datasets with different number of coexisting BANs
Dataset No. coexisting BANs Sampling Rate (per link)
Body-to-body links with different sensor locations LH– LH LH– RA LH– LW RH– RH RH– LW RH– RA RH– LH 1 8 BANs (3 transmitting) 66.7 Hz × × × × 2 8 BANs 25 Hz × × × 3 10 BANs 20 Hz × × × 4 20 BANs 10 Hz ×
[249] from different test statistics (i.e., difference between mean, variance, and distribution properties). Wide-sense-stationarity requires that the first and second moments (i.e., mean, variance, auto-covariance) of a time varying stochastic process
X(t) do not vary with respect to time t (Definition 5.1). In this chapter, WSS is tested over a wide range of window lengths (L) from 100 ms to 100 s, L = [100,200,300, ...,100000] ms. Here, we follow the process from [99], where the whole channel is divided into m consecutive non-overlapping intervals of length
` (where ` = L/2) to perform (m−1) independent pairwise comparisons across two consecutive intervals. Hence, for each window length L, i.e., (L = 2`), there will be (m−1) pairwise independent null hypothesis tests. An illustration of the pairwise comparison is provided in Fig. 5.2. We consider a lower bound for the amount of samples in an interval, i.e., 30 samples according to [250] to minimise the probability of Type-I and Type-II errors. We estimate the average probability of stationarity for a window length ofLover (m−1) tests with NHST, which provides the percentage of the channel being stationary as a function of L, disregarding the position of L over the channel. We consider:
H0: L retains WSS (null hypothesis).
H1: L does not retain WSS (alternative hypothesis).
Then,
pL=PTL> TLobs
Interval
(1) Interval(2) Interval(3) Interval(4) Interval(m−2) Interval(m−1) Interval(m) l L= 2l Window (1) Window (2) Window (m−1) Pc(1) Pc(2) Pc(3) Pc(m−2) Pc(m−1)
Figure 5.2: (m−1) pairwise comparison (Pc) across two consecutive intervals (`) wherepL(often called p-value) is a measure of evidence (i.e., strong evidence/weak evidence) against the null hypothesis2,3. This implies the observation of a more
extreme test statistic (TL) than the one actually observed (TLobs), given that the null hypothesis is true (observing a significant difference due to random sampling error while there was none or negligible difference).
if pL≥α, H0 is not rejected,
if pL < α, H0 is rejected in favour of H1,
whereαis the significance level/threshold for measuring the significance of the test outcome (based on pL). For instance, a pL≥0.05 indicates weak evidence against the null hypothesis as there is 5% or more risk of concluding that a difference exists when there is no actual difference, therefore the null hypothesis is not rejected. We examine different statistical significance levels (α) with α {0.01,0.05}, which corresponds to a confidence level (c`) of c` {0.99,0.95}, as c` = (1−α). For example,α= 0.05 implies that while there is 5% probability of incorrectly rejecting the null hypothesis, there is 95% probability that the confidence interval contains
2A measure of deviation from the actual outcome of the stationarity tests when the null
hypothesis is true. A higher value ofpL implies weak evidence against the null hypothesis and a
lower value implies strong evidence against the null hypothesis.
3The p-value is often interpreted as the probability of incorrectly rejecting the null hypothesis,
which is a misconception (as described in [251]) resulted from the mixing of two approaches (i.e., Fishers p-value approach and Neyman-Pearsons alpha level) in the widely used NHST.
5.3 Test of Significance for WSS 117
the null hypothesis value, i.e., 0 for difference, 1 for ratio [252].
The average probability of stationarity (γL) for a window length L over the entire period can be calculated as follows:
γL= m−1 P i=1 Wi m−1, Wi = 1, fpLi ≥α 0, fpL i < α, (5.4)
which indicates the percentage of pairwise comparisons (for a window length L) that satisfy the null hypothesis assumption over the whole period (from m− 1 pairwise comparisons). A given L with a higher value of γL, e.g., greater than or equal to 70% can be considered as WSS, as it fails to reject the null hypothesis for the majority of the cases. That also complies with the definition of WSS (second property of Definition 5.1), which is to have the WSS characteristics valid forLover the channel regardless of its position. When calculating the average pLi for the ith pairwise comparison (ithwindow) over multiple similar links from different subjects, we choose the median (typical) value (fpL
i
) to obtain a more robust estimation, as the median is not affected by outliers.
Here, we apply one-way ANOVA [253], Brown-Forsythe (B–F) [254] and Kolmogorov- Smirnov (K–S) [255] tests to evaluate the mean, variance, and distribution consis- tency of the body-centric channels. ANOVA relies on the assumption of the nor- mality and homogeneity of the variances of the underlying distribution. In general, the B2B channels are not normally distributed (they typically possess a skewed distribution). Fortunately, ANOVA is fairly robust to moderate deviations from normality [256], specially with a large number of observations. Additionally, it is not very sensitive against the homoscedasticity (homogeneity of the variances) as- sumption with balanced data (when the sets/intervals are the same size and have similar distribution) [257]. Alternatively, a non-parametric version of the ANOVA (Kruskal-Wallis (K–W) test [258]) can be used, which does not depend on the nor- mality assumption. By comparing the results of the K–W test and ANOVA test, negligible difference was observed. Hence, the classical one-way ANOVA analysis results are provided here.
deviation from the mean), which does not rely on the normality assumption, and therefore provides good robustness against many types of non-normal data while retaining good statistical power [260, 261]. Also, non-parametric tests are more useful when investigating physical phenomena, e.g., radio propagation, as unlike parametric tests they make no assumptions regarding the probability distributions of the sampled process [146]. The advantage of the K–S test (which tests whether the test samples come from the same distribution) is that the distribution of the test statistic does not depend on the underlying CDF (cumulative distribution function) being tested. The p-values for the test statistics are calculated using the asymptotic p-value calculation [262] with an approximation to the true distribution of the observed samples in each interval.
The description and estimation of the test statistics for these statistical hy- pothesis tests is provided in appendix A. We also investigated the variation in short-time power spectral coefficients [150] (appendix A) of the B2B channels in windowed data segments over time with dataset 3 in [247], where we estimated the variance of the multi-taper power spectral density (PSD) of specific data segments (e.g., 5s, 10s) over the whole channel. We found that for these data segments with most of the B2B channels, the power spectral variation is negligible, which satisfies the WSS assumption.
5.4
Experimental outcome of WSS investigation
In this section, we demonstrate the results obtained from the hypothesis tests with different experimental datasets (described in Section III and Table 5.1). We also provide some justifications of our findings based on the experimental results. The average probability of stationarity (found from ANOVA, B–F, and K–S hypothesis tests) with respect to different window lengths for dataset 1−4 (Table 5.1) is shown in Figs. 5.3 to 5.6, respectively. The probability of stationarity for a B2B link with a given sensor-location pair is estimated from (m−1) median p-values, calculated from (m−1) pairwise comparisons of multiple similar links between identical source and destination nodes.
5.4 Experimental outcome of WSS investigation 119